Details and Options

NDSolve[eqns,u[x],{x,xmin,xmax}] gives solutions for u[x] rather than for the function u itself.

Differential equations must be stated in terms of derivatives such as u'[x], obtained with D, not total derivatives obtained with Dt.

Partial differential equations may also be specified using the differential operators Grad (∇), Div (∇.), Laplacian (∇2), and Curl (∇). Typically these operators are used as in Inactive[op] to keep the operator form from evaluating.

NDSolve solves a wide range of ordinary differential equations as well as many partial differential equations.

The default setting of Automatic for MaxSteps estimates the maximum number of steps to be taken by NDSolve, depending on start and stop time and an estimate of the step size. Should this not be possible, a fixed number of steps is taken.

The setting for MaxStepFraction specifies the maximum step to be taken by NDSolve as a fraction of the range of values for each independent variable.

NDSolve typically solves differential equations by going through several different stages, depending on the type of equations. With Method->{s1->m1,s2->m2,…}, stage si is handled by method mi. The actual stages used and their order are determined by NDSolve, based on the problem to solve.